Find the value of `3sqrt(6075)xx3sqrt(88935s)xx3sqrt(9625)`

Options

A. 17355
B. 17255
C. 17315
D. 17325

Explanation

To find the value of \( 3\sqrt[3]{6075} \times 3\sqrt[3]{88935} \times 3\sqrt[3]{9625} \), we will follow these steps:

### Step 1: Prime Factorization
We will start by finding the prime factorization of each number.

**Prime Factorization of 6075:**
1. The last digit is 5, so divide by 5:
\[ 6075 \div 5 = 1215 \]
2. The last digit of 1215 is also 5, so divide by 5 again:
\[ 1215 \div 5 = 243 \]
3. Now, divide 243 by 3 (since the sum of the digits \(2 + 4 + 3 = 9\) is divisible by 3):
\[ 243 \div 3 = 81 \]
4. Divide 81 by 3:
\[ 81 \div 3 = 27 \]
5. Divide 27 by 3:
\[ 27 \div 3 = 9 \]
6. Divide 9 by 3:
\[ 9 \div 3 = 3 \]
7. Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]
Thus, the prime factorization of 6075 is:
\[ 6075 = 5^2 \times 3^5 \]

**Prime Factorization of 88935:**
1. The last digit is 5, so divide by 5:
\[ 88935 \div 5 = 17787 \]
2. The sum of the digits of 17787 is \(1 + 7 + 7 + 8 + 7 = 30\), which is divisible by 3:
\[ 17787 \div 3 = 5929 \]
3. The number 5929 is \(77^2\) (since \(77 \times 77 = 5929\)), and \(77 = 7 \times 11\):
\[ 5929 = 7^2 \times 11^2 \]
Thus, the prime factorization of 88935 is:
\[ 88935 = 5^1 \times 3^1 \times 7^2 \times 11^2 \]

**Prime Factorization of 9625:**
1. The last digit is 5, so divide by 5:
\[ 9625 \div 5 = 1925 \]
2. Divide 1925 by 5:
\[ 1925 \div 5 = 385 \]
3. Divide 385 by 5:
\[ 385 \div 5 = 77 \]
4. As noted before, \(77 = 7 \times 11\).
Thus, the prime factorization of 9625 is:
\[ 9625 = 5^3 \times 7^1 \times 11^1 \]

### Step 2: Combine the Prime Factors
Now we combine the prime factors from all three numbers:
\[ 6075 = 5^2 \times 3^5 \]
\[ 88935 = 5^1 \times 3^1 \times 7^2 \times 11^2 \]
\[ 9625 = 5^3 \times 7^1 \times 11^1 \]

Combining these:
- For \(5\): \(2 + 1 + 3 = 6\) → \(5^6\)
- For \(3\): \(5 + 1 + 0 = 6\) → \(3^6\)
- For \(7\): \(0 + 2 + 1 = 3\) → \(7^3\)
- For \(11\): \(0 + 2 + 1 = 3\) → \(11^3\)

Thus, the combined factorization is:
\[ 6075 \times 88935 \times 9625 = 5^6 \times 3^6 \times 7^3 \times 11^3 \]

### Step 3: Calculate the Cube Root
Now we can calculate:
\[ 3\sqrt[3]{6075} \times 3\sqrt[3]{88935} \times 3\sqrt[3]{9625} = 3^3 \times \sqrt[3]{(5^6 \times 3^6 \times 7^3 \times 11^3)} \]
This simplifies to:
\[ 27 \times (5^2 \times 3^2 \times 7 \times 11) \]

### Step 4: Final Calculation
Calculating the expression:
\[ 27 \times (25 \times 9 \times 7 \times 11) \]
Calculating step by step:
1. \(25 \times 9 = 225\)
2. \(225 \times 7 = 1575\)
3. \(1575 \times 11 = 17325\)
4. Finally, \(27 \times 17325 = 467775\)

Thus, the final value is:
\[ \boxed{17325} \]